We investigate the difficulty of finding economically efficient solutions to coordination problems on graphs. Our work focuses on two forms of coordination problem: pure-coordination games and anti-coordination games. We consider three objectives in the context of simple binary-action polymatrix games: (i) maximizing welfare, (ii) maximizing potential, and (iii) finding a welfare-maximizing Nash equilibrium. We introduce an intermediate, new graph-partition problem, termed Maximum Weighted Digraph Partition, which is of independent interest, and we provide a complexity dichotomy for it. This dichotomy, among other results, provides as a corollary a dichotomy for Objective (i) for general binary-action polymatrix games. In addition, it reveals that the complexity of achieving these objectives varies depending on the form of the coordination problem. Specifically, Objectives (i) and (ii) can be efficiently solved in pure-coordination games, but are NP-hard in anti-coordination games. Finally, we show that objective (iii) is NP-hard even for simple non-trivial pure-coordination games.

翻译：我们研究在图上寻找经济高效协调解决方案的难度。本工作聚焦于两种协调问题形式：纯协调博弈与反协调博弈。针对简单二元动作聚合博弈，我们考虑三个目标：(i)最大化社会福利，(ii)最大化势能，(iii)寻找社会福利最大化的纳什均衡。我们引入了一个具有独立研究价值的新型中间图分割问题——最大加权有向图分割，并给出了其复杂性二分定理。该二分定理作为推论之一，为一般二元动作聚合博弈中目标(i)的复杂性提供了二分结果。此外，研究揭示实现这些目标的复杂度随协调问题的形式而变化：目标(i)和(ii)在纯协调博弈中可高效求解，但在反协调博弈中属于NP难问题。最后，我们证明即便对于简单的非平凡纯协调博弈，目标(iii)仍为NP难问题。